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Friday, October 31st 2014

18:22

The Parable of the 1010 Minas

  • STATE OF EXISTENCE: Okay
  • IN THE BACKDROP: All is quiet on the Trick-or-Treat front.

Hover your mouse cursor over any number, and its decimal equivalent will pop up. Try it here: 111

It’s Reformation Day, the day on which we remember Martin Luther posting the 10101011111 theses on the door of the castle church in Wittenberg. Here in Hoosierland we celebrate the occasion by having children go from house to house demanding candy from the residents. Actually, lest anyone misunderstand my attempt at being clever, there is no tie-in between the Reformation and Halloween except that it happened to be on this day that Luther made his announcement. It was the day before All-Saints Day, which in English would be called "All-Hallows Eve," and somewhere the idea of ghosts and goblins and trick-or-treating arose, but it had nothing to do with Luther or the Pope. Actually, I don’t think that we’ll have much of a crowd on our front porch today because, as of this moment, it’s cold and rainy outside, and a drastic change in the weather  for the better is not expected for the next few hours.

This is the last part of the Monster Entry, the one in which we go back to Luke. You have probably noticed already that messing around with those number systems has confused my mind more than normally, and I keep accidentally switching between binary, dyadic, and decimal notations. So, please forgive me, and use what you learned in the previous three posts. Keep in mind that there are some overlaps between the systems. For example 1 is 1 in all three systems, 2 is 2 in decimal and dyadic, 3 is 11 in dyadic and binary. If you see a 3 or higher cipher written out, it must be a decimal number (since I’m limiting myself to these three systems). A 2 immediately rules out binary, and a 0 tells us that the number cannot be a dyadic one.

If this is freaking you out, just make use of the hover feature I have installed just for you.

Luke Bible Study Read the Text!

Bible Reading:
Luke 10011:1011-11020

vv. 1011: As they were listening to this, He went on to tell a parable because He was near Jerusalem, and they thought the kingdom of God was going to appear right away.

This parable is clearly very similar to the one recorded in Matthew 11001:1110-11110. Once again, it is true that all things are identical if you ignore their differences, but there are differences, and we should think of them as parables that are alike, but not the same. To be sure, both parables come down to the message that we must be faithful to Christ while we wait for his return, but the setting in Luke also carries political Jesus reality check.

I must tell you that in this long time of working through Luke, I have been surprised by the number of parables whose fundamental interpretation is already given in the text, either by Jesus in reciting it or by Luke in his introductions. In this case, Luke helps us understand the meaning of the parable by noting that Jesus told it in response to his disciples’ expectation that the time had come for him to set up the kingdom of God. We’ll come back to that point presently.

First, though, we need to recognize the fact that Jesus alluded to some real historical events in the story. This observation does not mean that the entire parable is supposed to be a historical narrative; including some historical allusions does not mean that the parable isn’t still a parable. Nevertheless, the way in which Jesus framed it would have struck a chord of recognition in his audience. When King Herod the Great (the baby killer) died in 12 BC, it had only been a few weeks after he had changed his last will to the effect that he named his son Archelaus as his successor. Archelaus was known as a cruel and harsh man, and the populace was afraid of having him as their king, preferring his younger brother Antipas. Archelaus had to travel to Rome in order to have his kingship confirmed by the emperor; he was followed by Antipas, their half-brother Philip, and other messengers opposing his rule. Nonetheless, the emperor Augustus confirmed Archelaus’ position over Judea, Samaria, and Perea, though he did not permit him to carry the title “king.” Instead, he was only allowed to be called “tetrarch,” which means ruler of a quarter of a kingdom, which put him on a par with Antipas, who governed Galilee, and Philip, who received territory further north. Apparently the title “tetrarch” was appropriate even when the land was divided among 11 rulers rather than 100. When Archelaus returned home,  he vented his anger on those who had opposed him within his allocated land by staging a blood bath.

It is important to realize that the close-out of the parable in v. 11011, “But bring here these enemies of mine, who did not want me to rule over them, and slaughter them in my presence,” should not be attributed to God in some sort of analogical fashion, but that they are something that the evil ruler in the parable says. In Jesus’ parable, the man who returned  had become king, though, as we said, Archelaus did not have that title.

There were diverse opinions among first-century Jews as to the attributes of the coming messiah, and not everyone even expected one. But in general, the anticipations included a political mission for the messiah. He would reestablish the Kingdom of David. Thus, as Jesus was heading up to Jerusalem, accompanied by a large parade of people, many of them thought that they were about to witness how Jesus would expel the Romans and take over as the true king.

The details of the parable are fairly straightforward. Before the would-be king left, he gave each of his servants 1010 minas, a unit of weight and currency (weight of the precious metal), whose actual value changed from time to time. It seems to have hovered around 60 shekels, which does not tell you much unless you know that a shekel weighed 211 grams or a little more. The would-be king expected his servants to use that money wisely on his behalf and hopefully make a profit in the process.

When the ruler returned as king, the first servant announced that he had made another 122 minas, and the ruler gave him authority over 1010 towns as a reward. Clearly the realistic basis of the parable ends here. The second servant earned 21 minas, and the king, displaying a fondness for symmetry, put him in charge of 101 towns. A third servant did nothing with the money entrusted to him. In fact, he made use of the occasion to let the ruler know what a bully he was. This man apparently was not very smart. First of all, he should have tried to put the money to work in some way, even if it was nothing more than to put it into a savings account and accumulate interest. Second, given the fact that the ruler was unjust and violent, it would have been best if he had not thrown this accusation at him. The king consigned him among those people who would be executed.

Jesus spoke this parable in light of the fact that a number of his followers expected him to reestablish the kingdom in just a short while. So, how does the parable respond to those hopes? I see 10 points here.

1. Those who belonged to Jesus should be prepared for a lengthy interval before Jesus would actually return as king. It is simply not true that everyone in the early church was living in the anticipation that Jesus would return in a very short time. He could, but he also might not. Please remember that Jesus spoke these words, but that early Christians recorded them and would have had this message right in front of them.

10. During the time of waiting, we should remain faithful to whatever God has called us to be and do. I need to make something clear here. There is no threat to Christians in this parable. The allegiance of the unfaithful servant was against the king, and a Christian’s allegiance is not against Christ. It may take a while yet, but when Christ returns, our standing in him is going to be an occasion for celebration, thanks to his grace.

 

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Wednesday, October 29th 2014

15:39

Numbers, part 3

  • STATE OF EXISTENCE: pooped

I'm continuing the monster entry. Please read the preceding two entries before this one, or nothing will make sense. My next step needs to be to bring this all together in a meaningful way, but not right now. Once again, here is the index to connect things somewhat.

Concession Stand Math Numbers, pt. 1
Clapping on 3 Numbers, pt. 1
Binary Numbers Numbers, pt. 1
My Age in Binary in Numbers, pt. 1
Binary Continued Numbers, pt. 2

2014 in Binary Numbers, pt. 2
Dyadic Numbers Numbers, pt. 3
My Age in Dyadic Numbers, pt. 3
Dyadic Chaining in Numbers, pt. 3
The Parable of the 122 Minas--Coming up!

IV. Dyadic Numbers. Raymond Smullyan has invented an adaptation of the binary system, which he finds more useful for various mathematical reasons. He calls it dyadic notation. It works very much like the binary system, except that it uses 1s and 2s instead of 0s and 1s. The dyadic system, just like the binary system, can express any positive integer, but it does not allow us to write 0.

Again, we have a series of boxes or placeholders, and again, they represent the powers of 2, but each one contains either the designated multiple of 2 or double that amount. There are no empty boxes.

               

PLACES

27

26

25

24

23

22

21

20

2n or

2 x 2n

128 or 256

64 or 128

32 or 64

16 or 32

8 or 16

4 or 8

2 or 4

1 or 2

POSSIBLE VALUES


So, in the far right position we can have either a 1 (20) or a 2 (2 x 20). Counting upwards, you might be tempted to write a “2” as 10 (which would be correct for binary numbers), but, as we said, just as there is no crying in baseball, there is no 0 among the dyadic numbers. A 2 shows up as a “2” in the rightmost box this time (2 x 20).

Under this system, we need to shift boxes to the left whenever the number would be a 3 or higher. So, when we want to write the decimal 3 in dyadic, we can’t put a 3 into the final position, and we must move forward. The dyadic version of 3 is 11: 1 in the 21 (=2) position, and 1 in the 20 position (=1). 4 is 21+ 2 x 1, and we can write it as 12 (one 2 and two 1s). Here are the first twenty integers in Smullyan dyadic style:

Decimal

Dyadic

 

Decimal

Dyadic

 

Decimal

Dyadic

 

Decimal

Dyadic

1 =

1

 

6 =

22

 

11 =

211

 

16 =

1112

2 =

2

 

7 =

111

 

12 =

212

 

17 =

1121

3 =

11

 

8 =

112

 

13 =

221

 

18 =

1122

4 =

12

 

9 =

121

 

14=

222

 

19 =

1211

5 =

21

 

10 =

122

 

15=

1111

 

20 =

1212

Some of the dyadic numbers are the same as their binary counterparts, but constructing them calls for a very different kind of procedure. In binary code, if you want to express the decimal 16, you can just plunk a 1 into the 16-spot and fill out the rest with 0s. But since the dyadic code does not have 0s, that’s not possible. No location can go entirely unfilled until the number is completed. Again, it must get either one or two instances of 2n, as called for by the spot in question. Dyadic numbers can be read as easily as binary ones, but to construct them properly involves a little bit of trial and error, at least for me.

So, if I want to write out my age (65) in dyadic, I can’t just put a 1 into the place for 64, bring in lots of 0s, and then finish with a 1 at the end. In fact, I won’t even need the slot for 64. For binary numbers it’s usually easiest to start from the left. For dyadic numbers we need to do a little more manipulating to make sure that each place has either a 1 or a 2. There are no empty placeholders, and so we may need to do a little planning ahead. If someone reading this discussion has an idea on how to make this process automatic and mechanical, please share it with the rest of us.  

Since the number 65 is odd, we know that it will need a 1 as the last digit, which stands for 20, which, as we said, equals 1.

32

16

8

4

2

1

?

?

?

?

?

1

 

That leaves 64 units to distribute over the row of boxes that we started with 32 on the left. We can do so by allocating them in this way. As mentioned, I arrived at this number by juggling them, so there is no methodology that I know of to walk us through.

32

16

8

4

2

1

1

1

1

1

2

1

111121

To write out the decimal 65 in dyadic, each slot wound up containing one representative, except for the one designated for 2s, where there are two 2s. You might note that, if we were to read the binary and dyadic numbers as decimals, I would be roughly 900,000 years younger in the dyadic format. But that's nonsense and unhelpful, so it's great fun to indulge in. Nevertheless, we must proceed.

The obvious question comes up: What can you do with this? It seems that there is not a whole lot of practical application for grocery shopping or playing music, though dyadic numbers figure prominently in contemporary presentations of arguments leading up to the Gödel Undecidability Principle. But that's something people talk about a whole lot more than use. However, there is a "mathematical" operation you can do with dyadic numbers, which turns out to be fairly intriguing.

Please, if you're sick to death of this math stuff already, skip this next section and wait for the next entry. I don't intend to punish anyone. Of course, if deep down you feel you need to do penance for something, and you consider reading about math penance, by all means, put on the numerical hairshirt.

I'm continuing to give you some highlights out of Smullyan's book, A Beginner's Guide to Mathematical Logic.

What you can do with dyadic numbers is add them up. -- Well, it's not really addition, except we're just literally putting two dyadic numbers together. Smullyan calls it concatenation; we can stick with the English equivalent, "chaining together." Let us conjoin the dyadic versions of 6 and 7. The symbol Smullyan uses for concatenation is an asterisk: *. So, our problem is 6 * 7 (in dyadic) = ?  The dyadic equivalents are: 6 = 22; 7 = 111. Then

22 * 111 = 22111

That's right, we just pasted the two number together and got what appears to be nonsense.

22111 in decimal translates into two 16s, two 8s, and one each of 4, 2, and 1. Then

32 + 16 + 4 + 2 + 1 = 55

and consequently, in decimals again,

6 * 7 = 55

So, in dyadic chaining, if we call the first number x and the second number y,

x * y = "x glued to y" or x y,

and we still want to know what in the world one could possibly find of interest in this apparently meaningless application of our mental super glue.

Yes, there is something of interest, at least if you're fascinated by the way that numbers behave. It might help to know that Smullyan started out his career as a stage magician before turning to mathematics and Daoist philosophy. (Thanks to nephew Michael for pointing out the latter fact to me.) So, let's try some Smullyan magic. By the way, he cautions us to keep in mind that in this case x y does not stand for x multiplied by y (as in xy); that's why I put a space between the two variables.

We are going to start with a really weird move, namely to count the number of ciphers in the dyadic number represented by y. In our example, the number 7 is translated as 111, giving us 3 ciphers, which we're going to designates as L, the Length of the number. Thus,

L of y = 3

L(y) = 3

Now we use L as an exponent for our steady companion, the number 2, and create 2L. In this example, since L = 3,  2L is 23 or 8.

Then we multiply 23 by x, which was 6. So 8 x 6 = 48.

Does this result do anything for us? Can you see anything in that number 48 that could raise our interest?

Let's go back to that very strange number 55, which was the result of chaining 6 and 7. We can ask ourselves, "Is there any kind of significant relationship between 48 and 55?" Something pops up. "What number do we get if we subtract 48 from 55?" The answer is, of course, 7. Now, do you recall the importance of the number 7 in our example? Of course, you do. It's the number that we called y.

Then we get the following equation:

6 x 23 + 7 = 55

So, the general equation for a result of chaining two dyadic numbers together is:

x multiplied by 2L plus y = x and y chained together, or

x x 2L + y = x y

Do you want to try it out with a different set of numbers? Here we go:

3 * 16 = 3 chained to 16

11* 1112 = 111112

We translate the resulting dyadic number:

Dyadic 111112 = Decimal 32 + 16 + 8 + 4 + 2 + 2 = 64.

This time y is 4 digits in Length, so

L(y) = 4

Again the magical formula:

x x 2L + y = x y

Then, in this case:

3 x 24 + 16 =

3 x 16 + 16 =

48 + 16 =

64.

And, of course, 64 (111112) was our result of chaining 3 (11) and 16 (1112) together. Do you thing this is cool? I do.

Just for fun, try some numbers of your own.

I must admit that I'm a little pooped from writing all of this out without wanting to make a mistake or skip a crucial step, so I'm going to wait until the next entry to get back to Luke and the parable.

Not-so-by-the-way, I recommend Smullyan's book. As I intimated above, for a mathematical genius, such as Smullyan, the concept of a "beginner" is often slightly different than for the rest of us. Then again, Smullyan tells us that understanding it may require multiple readings of various passa

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Wednesday, October 29th 2014

12:03

Numbers, part 2

  • STATE OF EXISTENCE: Not a 0

I need to pick up right where the character limit to a blog entry cut me off last night. If you have not read the previous entry, please do so. This won't make a lick of sense without it. Also, here is the index to the various parts.

Concession Stand Math Numbers, pt. 1
Clapping on 3 Numbers, pt. 1
Binary Numbers Numbers, pt. 1
My Age in Binary in Numbers, pt. 1
Binary Continued Numbers, pt. 2

2014 in Binary Numbers, pt. 2
Dyadic Numbers Numbers, pt. 3
My Age in Dyadic Numbers, pt. 3
Dyadic Chaining in Numbers, pt. 3
The Parable of the 122 Minas--Coming up!

I was discussing binary numbers in preparation for dyadic numbers. In our decimal system, whenever a number in a certain place adds up to more than 9, we move one spot to the left and put in the appropriate digit there. Thus we write 9 as a single digit, but 15 has two digits, where the digit to the left does not stand for 1s but for 10s. With binary numbers, you shift to the left for every time that a 2 comes up. If a 2 belongs into a position, you mark that next spot with a 1. If there is nothing there, you put in a 0. I showed you in the last entry that my age in binary style is

1000001

Let’s now go in the other direction. Say you have won 37 games of "Words with Friends,"and we want to express that number in binary code. We need to reserve enough places that there is room for a 32 as our largest number. We could put a whole bunch of 0’s to the left, but that’s unnecessary because clearly we don't need a 64 or anything higher.

32

16

8

4

2

1

?

?

?

?

?

?

37 is larger than 32, so we can allocate a 1 in the 32 box, subtract 32 from 37, and have 5 left to allocate.

32

16

8

4

2

1

1

?

?

?

?

?

Then, adding a 16 or an 8 would give us a higher number than 37, so they each get 0’s.

32

16

8

4

2

1

1

0

0

?

?

?

Is there room for a 4? Yes; we can put a 1 into the place for 4’s.

32

16

8

4

2

1

1

0

0

1

?

?

 

We have now created our binary number up to 36 (32 + 4), just 1 short of the goal. If we were to put a 1 into the very next spot to the right, that would mean that we were adding a 2, which would give us 38, again too large. So the location for 2’s gets another 0. Now we are looking at

32

16

8

4

2

1

1

0

0

1

0

?

and clearly the last place gets a 1, representing the value of 1.

32

16

8

4

2

1

1

0

0

1

0

1

Thus we have the full binary notation for the decimal number 37:  100101.

One other example. This is the year AD 2014. To write this number in binary style, we need to draw a few more boxes. 1, 2, 4, 6, 8, 16, 32, 64, 128, 256, 512, and 1024. The next higher multiple of 22048, cannot be of any use to us since it exceeds the decimal number we want to express.

1024

512

256

128

64

32

16

8

4

2

1

?

?

?

?

?

?

?

?

?

?

?

 

We put a 1 into the 1024 place and subtract 1024 from 2014,

1024

512

256

128

64

32

16

8

4

2

1

1

?

?

?

?

?

?

?

?

?

?

which leaves us with 990 to fit in. There is plenty of room for 512, so that box gets a 1 as well,

1024

512

256

128

64

32

16

8

4

2

1

1

1

?

?

?

?

?

?

?

?

?

and we still have 478. We can also place a 1 into the box for 256 and subtract that amount from 478.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

?

?

?

?

?

?

?

?

Then we still have 222 to allocate. 222 can accommodate 128, so let’s put a 1 into that location and subtract it from 222.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

?

?

?

?

?

?

?

The remainder is 94. We can add yet another 1 because clearly 64 contributes to the sum.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

1

?

?

?

?

?

?

 

94 – 64 = 30. Finally we get to write a 0 because 30 is smaller than 32, and, thus, we cannot include 32.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

1

0

?

?

?

?

?

But 30 definitely leaves lots of room for 16, so we subtract 16 from 30 and have 14, with the 16 spot having earned its 1.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

1

0

1

?

?

?

?

 

14 contains 8, leaving 6. 8 gets a 1.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

1

0

1

1

?

?

?

6 is the sum of 4 and 2, so each of those two places get a 1 because they are represented.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

1

0

1

1

1

1

?

These figures add up to 2014. Since that's an even number we could have marked the 0 in the last spot right at the outset.

1024

512

256

128

64

32

16

8

4

2

1

1

1

1

1

1

0

1

1

1

1

0

Expressed in binary code, we live in the year

AD 11111011110.

Now, I wouldn't have gone through all of this if I didn't want to take a further step and tell you about what I learned from Smullyan's book about dyadic numbers, but that will have to wait. Then I will also talk about a parable that happens to include some numbers.

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Wednesday, October 29th 2014

1:15

Numbers, part 1

  • STATE OF EXISTENCE: seeing 1's and 0's when I close my eyes

This is the first part of the monster entry I was warning you about. I have postponed some sections and provided links at the start here so that you can skip over whatever doesn’t interest you. I'll also post a collected version, in which you don't have to read entries backwards. 

Concession Stand Math Numbers, pt. 1
Clapping on 3 Numbers, pt. 1
Binary Numbers Numbers, pt. 1
My Age in Binary in Numbers, pt. 1
Binary Continued Numbers, pt. 2

2014 in Binary Numbers, pt. 2
Dyadic Numbers Numbers, pt. 3
My Age in Dyadic Numbers, pt. 3
Dyadic Chaining in Numbers, pt. 3
The Parable of the 122 Minas--Coming up!

 

So while we were gone last week, what did I do when we weren’t out and about? For one thing, I was thinking about numbers, biblical and otherwise.

We’ll start with the “otherwise.” In this case, as you will see further below, “otherwise” consists of various disconnected trains of thought, presented for your entertainment purposes only--or at least that's the impression I wish to create.

I. Concession Stand Math. The Grand Ole Opry building has concession stands in the lobby, much like a sports facility. On Tuesday evening of last week, I had brought a small bottle of water along; June had forgotten hers. I asked her if she wanted anything from the concession, and she said that a bottle of water would be good. (I would have gladly shared mine, but it really was small, and there wasn’t that much left.) So I stood in line watching the pre-show inanities on the screen, waiting for it to be my turn. While standing there I realized that I was actually a little hungry. Oftentimes on trips we eat a big meal late in the afternoon (you can call it “Tea,” if you like), which pretty much carries us through, except for the occasional snack. Well, I decided I wanted a hot dog (technically an “Opry Dog”). I noticed on the price list that one of those cost $4.25, a confiscatory price even for a fairly large wiener, but in keeping with the time-honored marketing approach of fleecing tourists. Large drinks were $4.00. Concession MathThe real bargain came into play If you were to buy both, a hot dog and a drink you could get them at the astounding rock bottom price of $8.00, saving 25 cents. So, I got $8.00 ready, but had a quarter easily available as well, just in case I was missing the fine print. When it was my turn I asked for a hot dog and a bottle of water.

“That’ll be $8.25,” the young lady stated immediately.

“Oh, the special price does not apply when you get water?” I asked, placing the quarter on the pile of notes I had set on the counter.

“Hm, let me check; I think that’s only when you get chips as well.” (For my readers from the Commonwealth, in American English “chips” are potato chips, not French Fries, as we call les pommes frites here.

“Okay, then, I’ll have a bag of chips as well.”  

She went to the other side of the stall, talked to someone, and returned with a small bag of potato chips. Somewhat apologetically she said, “I’m afraid that’s actually $9.00 now.”

What’s the use of arguing? I was there to have fun, and she was obviously not caught up or not trained or—most likely—neither. I removed the quarter, added a dollar bill, and thanked her for her service as I walked away with the imitations of nutrition, trying to figure out the equation behind this transaction.

No Clap SignII. “Friends don’t let friends clap on one or three.” Someone posted that slogan on Facebook a little while ago, and I was glad to see it. Over the years I’ve tried to get the occasional audience to see the point. Here is what it means: When most American audiences clap along to a catchy song, they usually clap on the first beat of a bar along this pattern:

clap-2 / clap-2  / clap-2   ...

To be honest, it can drive me crazy at times.  Clapping on the second beat actually gets you much more into the song.

1-clap / 1-clap / 1-clap  ...

I remember once wondering for a moment why I found it so much easier to stand and clap along with the music in an African-American church; then I realized that they were clapping on 2, not on 1. Just this last Saturday (actually, now week before last), at Cowboy Church, I reminded the good folks there to follow this pattern. 1-clap, 1-clap, 1-clap, as I was singing “Just a Little Talk with Jesus” in a somewhat jazzed-up style, and I think they caught on to the feeling.

But there is one strange exception, which is provided by old-fashioned rock and roll. For example, I noticed it on one of the song Keith Urban was singing the other night. For good old rock, the emphasis goes often goes on 3, i. e. the third beat. Now that may sound really strange, if not impossible, to you, but there’s a trick. If you were to look at the music written out, you would see that you’re still clapping on the first beat of a measure. However, it would be preceded by two beats in the previous measure. To really simplify the thing, a stereotypical rock number begins with two prior to the four beats to the first full measure.

Let’s put it this way and distinguish between pauses (P), light beats (L), and heavy beats (H). Then the pattern looks like this:

P-P-(L-L / H-L)-(L-L/ H-L)-(L-L / H-L)-L-L

The slashes stand for the break between bars, and the parentheses indicate the rhythmic feel that the musician actually conveys to the audience.

III. Binary Numbers. I’ve been reading the very colorful Raymond Smullyan’s Beginner’s Guide to Mathematical Logic. I’m sure that, when (or if) he reads this, he will get a kick of my posting a black-and white picture of him after describing him as “colorful.”

Raymond SmullyanAt one point in his exposition, which “beginners” will find somewhat challenging as a whole, he brings up the idea of dyadic numbers. They are very similar to binary numbers, but carry an interesting twist, and that's one part of this monster entry I'm postponing. First let’s take a look at or review, as the case may be, the binary system. The contrast will hopefully also highlight what’s so good about our decimal system. Presumably, one could count without any system, but then every number on the infinite number line would have to have its distinctive symbols with no repetitions and no intrinsic ordering in what we write down. As wearisome as Roman numerals can be, even they have a pattern. (The Romans were extremely intelligent, by the way; even their youngest children learned Latin at a very early age.) To get back to the point, as an example, the decimal numbers 4, 5, and 6 could theoretically be written as iiii, iiiii, iiiiii, but instead, the Romans wrote iv, v, and vi. Still, the set-up was quite cumbersome, particularly for anyone who would actually do math with them, which was the case right through the Middle ages.

So, the ancient Hindus came up with a system in which a number consists of serials of numerals (the symbolic representations), each of whose value is partially determined by its place in the series. The function of 0 (zero) as a placeholder greatly aided the construction of this system. Imagine a set of boxes representing place values. Conceivably they could stretch out toward infinity toward the left, but I don’t want to use up all my bandwidth for that purpose. With no other values indicated, theoretically there are implicit 0s in all places.

0

0

0

0

0

Now, take the numeral 7; put it into the first box on the right, you have the number 7.

0

0

0

0

7

Stick it into the second box, and it means “70.”

0

0

0

7

0

You don't need me to teach you how to write numbers; I'm only setting this up to make the contrast with the binaries. Each box can contain any other number from 0 to 9. Let's leave out the unnecessary 0s from now on.

Can we put a 10 or anything larger into any box, such as the first one on the right?

 

 

 

10

No, as soon as we get up to 10 or more we need to put the excess multiple of 10 into the appropriate box on the left.

 

 

 

1

0

So, if we add 7 + 7, we don’t get  --

 

 

 

 

14

but

 

 

 

1

4

Each time the value of a place exceeds 9, we need to push the number one space further to the left. I apologize for this obvious stuff, but I like to set up the easy things so that the more difficult ones won't be as confusing.

So, let's now imagine that we don’t wait for 10 to shift to a new box on the left. Instead, we move to the left every time the number would be 2 or more. Then we are working in the binary system. As we all know, I imagine, this is the system according to which computers and many other digital devices function. There are only two ciphers, 0 and 1.

In that case, the boxes now don’t represent numerals up to nine, but powers of 2. Here are a few numerical facts.

We all know that 22 means 2 squared (2 x 2), i.e. 4, and 23 is 2 cubed (2 x 2 x 2), namely 8.

Let me just mention that any number raised to the first power, e.g., 21, 71, or 431 equals the number itself, viz. 2, 7, or 43.

Finally, any number raised to the "zeroth" power, e.g., 20, 70, or 430 equals 1.

So now if every new exponential power of 2 needs to go into a new place to the left, a 1 in a box means that there is one 2n living there. A 0, of course, means it's empty, but the 0 is important as a place holder. There may be 1s before or after it.

0

0

0

0

0

0

0

0

PLACES

27

26

25

24

23

22

21

20

2n

128

64

32

16

8

4

2

1

MAX VALUE

So, let’s set out to make the comparison. We'll drop the boxes and the helps as we go along.

The number written out as 0 in a decimal system would be written out in binary as 0.

0

0

0

0

0

0

0

0

 

The number 1 is represented as 1.

0

0

0

0

0

0

0

1

But it’s not time to yawn; from here on out it gets a little more interesting because the decimal number 2 cannot be written as 2 in the binary system, just as the number 10 cannot be written in a single space in the decimal system. The binary system has no symbol for 2 other than its placement that's marked with a 1. The second box from the right is reserved for 2s. Thus we place a 1 there, meaning that there is one 2 in that place. The first box is empty and marked with a 0.

0

0

0

0

0

0

1

0

Of course, that's usually written without the boxes simply as 10. So the decimal 2 is the binary 10. Quite a difference if you forget which system you're in!

We need to come back to the first box in order to write the number 3 which comes out as 11.

0

0

0

0

0

0

1

1

In fact, all odd numbers will need a 1 in the first box on the right since all of the other boxes contain multiples of 2, making them even if they are occupied. Here are all the binary numbers up to decimal 20.

Decimal

Binary

  Decimal Binary   Decimal Binary   Decimal Binary

0 =

0

 

6

110

 

11

1011

 

16

10000

1 =

1

 

7

111

 

12

1100

 

17

10001

2 =

10

 

8

1000

 

13

1101

 

18

10010

3 =

11

 

9

1001

 

14

1110

 

19

10011

4 =

100

 

10

1010

 

15

1111

 

20

10100

5 =

101

   

So, as long as we keep in mind which multiple of 2 is represented in a particular place, we can convert binary numbers into decimal numbers just by adding. Sum up the 2ns in keeping with their proper location, don’t forget about the 1 on the furthest end on the right, and you get the decimal equivalent.

In binary form, I’m 1000001 years old.

Start from the right of the number and count up the powers of 2: 0, 1, 2, 4, 8, 16, 32, 64 from left to right.

64

32

16

8

4

2

1

1

0

0

0

0

0

1

There is a 1 in the left-most box we have, which is the one for 64 (25, 1000000 in binary), and one on the extreme right, where the value is 20 (= 1). Add 64 and 1, and you know that in decimal notation my age is 65

I just tried to post all of the first part of this entry and obviously exceeded the allocated number of characters. So, I need to leave you hanging earlier than I anticipated, and this monster will probably need to be slain into three parts rather than the two I had anticipated.

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Monday, October 27th 2014

19:55

Country Music Classic

  • STATE OF EXISTENCE: good

Larry Gatlin and the Ryman Staff BandI’m still working on what was supposed to have been a “parallel” entry last Tuesday or somewhere around that time. By now it has turned into a monster of six pages and is not yet done; I’ll have to break it up into parts whenever I finish it.

In the meantime, here are some more pictures and dispersed comments on the country music experience in Nashville. Thursday evening (of last week by now) June and I went to the show in the Ryman Auditorium called “Country Classics,” a spin-off of the Opry, playing the kind of music that I would readily identify as country. It follows the Opry format and is also broadcast on 650 WSM. The host for the show was the irascible Larry Gatlin. Along with him came brother Steve playing the bass and daughter Kristin, who sang back-up with a gorgeous low alto voice, and, of course, the staff band. Larry told us that brother Rudy was in jail, and he related a hilarious story of how that supposedly happened, but I doubt its veracity.

The first half followed the traditional opry format. Larry as the host sang a few songs, then introduced John Conlee, whose voice still has the power of a large organ pipe. It was a treat when for his second song he put on his rose-colored glasses and sang his hit by that title. Then Larry introduced a newcomer, Ashley Monroe, whose voice has the timber of, say, Dollie Parton, but not yet the power and assurance that comes partially with experience.The next person on the program was also someone new—to me at least. Shawn Camp did a few songs in early-seventies country style, and he brought it off well. If I may be allowed another analogy, Merle Haggard’s style comes to mind.

John ConleeAshley MonroeShawn Camp

I’m ignoring Larry Gatlin’s off-and-on antics in between the acts and while introducing them. He kept us in stitches. Not only did he say good things about each performer, he made his compliments specific enough that you knew he wasn’t just repeating standard lines. At the end of the first hour Larry, Steve, and Kristin sang a number of gospel songs. If you know anything about Larry, his personal highs and his lows, you know that when he sings gospel, it comes from his heart. He encouraged everyone to sing along, which I did, making up harmonies as well.

Steve GatlinLarry Gatlin and his daughter Kristin

Larry Gatlin All the GoldSo, dare I say the following? – Umm. – I’ll risk it. The lady who sat to the left of June who, in turn, sat to the left of me, mentioned to June that she thought my singing was as good as that of the performers, and that I could just as easily be down on that stage. Okay, sorry to have brought that up, but it was a nice compliment, and if I can’t share a compliment with you, my faithful readers, who else could I tell about it? – Of course at the time I just said “I know.” – No, I didn’t. I was suitably humble.

To open up the second half, Larry sang his traditional closing number, “All the Gold in California.” (That’s something that I always used to do when I did performances on the church basement circuit. I would do my closing song right after the first song so that I knew that pretty much everyone was there until the end. Then I would proceed with the rest of my program.) As always, Larry made “All the Gold” an audience-sing-along. It was, indeed, his last song of the evening. Then he turned over the rest of the program to Lorrie Morgan, who once upon a time was known as the daughter of George Morgan, a famous country musician in his day. Over the years she has very much carved out her own niche, and I suspect George Morgan is now primarily known as the father of Lorrie. She had her own back-up group, including her daughter-in-law, whose name I didn’t catch, playing guitar and providing vocal harmonies. Her bass player finally convinced me of something I had suspected all along, namely that bass players, including your bloggist, are somewhat eccentric as a class.

Lorrie Morgan Lorrie Morgan

Lorrie Morgan's Bass Player Larry Gatlin and Lorrie Morgan

If you want to see and hear country music as most of us remember it, the Country Classics show at the Ryman is the place to go.

For our final musical treat, on Friday evening we went on the three-hour cruise of the General Jackson, a sort of traditional showboat with paddle wheels. (Some pictures may be forthcoming shortly.) It was the kind of thing that was fun to have done once, but on the whole not worth the money, particularly since they wound up charging extra for all kinds of things, such as soda pop to go with the surprisingly small dinner buffet. The show, “The Heart of Tennessee,” was fairly well done, though the young ladies’ Tina Turner style costumes grew old after a while. On the whole, as I said, it wasn’t bad, but we could have done better things with what that experience cost.

On Saturday we drove home. It’s not a long trip between Nashville and here in Smalltown, USA, but I was pretty drained. June suggested several times that we could stop over somewhere along the line, but I really didn’t want to, and we got home fine. I did, however, go to sleep for the night at six in the evening, something that’s virtually unheard of. I was awake for a little while in the middle of the night, but went right back to sleep.

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Wednesday, October 22nd 2014

21:53

Tuesday Opry

  • STATE OF EXISTENCE: musical

Believe it or not, I really wanted to post two entries tonight, this one on the Opry Concert and a conceptual one. Obviously, I underestimated how long it would take to prepare these pictures and get them properly situated.

Last night June and I went to the Grand Ole Opry house, just outside of Nashville, for "Tuesday Night Opry," something we had not yet experienced, having always been there on either a Friday or Saturday night. On the weekends, one usually gets a number of the retirement-aged stars of a while back, along with hosts such as Porter Wagoner or Lorrie Morgan—and, of course, one or two current chart toppers. Tuesday night not so much. The Grand Ole Opry is actually a vastly elaborated live radio show on WSN, and last night the radio announcer, not an active country musician, was the official host. The acts were mixed in style and quality. Each one got to do three pieces, except for the main attraction at the end of the show, who quite properly got to do twice as many.

Diamond RioA Leitmotif of this entry is to give a quick assessment of the question where "country" music is heading. Please don’t misunderstand; in my book you don’t have to be a country singer to be an outstanding singer/musician. My recent ravings about the BeeGees, CSN, and Billy Joel should have made that clear. It’s just that I hate to see the country genre disappear. I realize that my approval is neither highly sought for, nor should it be. But if I didn't express my opinions on my blog, I wouldn't have anything to write for either my own amusement or that of my readers, who are undoubtedly already finely attuned to sorting out the wheat from the chaff of my pronouncements.

I have at least one picture for each performer. From where we sat, using a telephoto lens provided the only means of getting any worthwhile photos at all, but the lighting was so low that most pictures did not turn out--not that I expected them to. Fortunately, with digital cameras, you can check and make adjustments as you go. So, by the time of the intermission, to my pleasant surprise, I had figured out that if I steadied my camera as best as I could and took a series of shots in a row, by sheer probability at least one of those shots would be fairly close to half-decent focus. God bless digital cameras! Still, there was no point in adding larger versions for most of the pictures here since I had stretched the possibility of getting a sharp image for most of them as far as they would go. I’ll mention the exceptions below.

First up was Diamond Rio, a good group to open such a program. They sounded amazingly upbeat and fresh. When they first came on the scene in 1991 (their very first recording was a #1 hit), others shared my view that they were pushing the limit of what should be called "country.” At this point, in contrast to more contemporary “country musicians,” they appear to be traditionalists.

Native Run

Next came Native Run, Rachel Beauregard and Bryan Dawley, whose mentioned credits include opening up for a number of well-known country artists. I would suspect that they may have a lengthy career as an opening act. Their sound pretty much tended toward the rock column. The only good picture I got of them was was one without the telephoto lens, but it captured the entire set-up in all its cluttered magnificence, and you must click on it so that you will get a full view of the entire Opry. This mandate is particularly important if you've never been there, but if you have, you still should because it's a cool picture that I think you will like. Note that huge screen showing Rachel in action.

 

Chase Bryant

Speaking of the loss of the “country” in “country music,” the next budding young artist on stage, Chase Bryant, had no recognizable “country” sound (at least to me) in his performance. My assessment was that Roy Acuff would have turned in his grave if he had heard these sounds in the Opry. I guess the country label moguls must know what the market is looking for, but to me it’s sad to see the Grand Ole Opry become the home of “market music.” There’s a place for it, but why does it have to swallow up its roots? Anyway, Chase has written some interesting pieces, and he has a clear voice. He has yet to find his niche, though I'm afraid his back-up group with their exaggerated contortions didn’t help his cause all that much either. Then again, for all that I know he may win twenty Grammies before his career is done. We curmudgeons do no have a vote.

Closing out the first half was the Grammy-winning blue grass group that calls itself the Del McCoury Band. Del M. has performed at the Opry for five decades, first as a member of Bill Monroe’s group and now heading up his own ensemble. Every one of his musicians, including his sons (mandolin and banjo), have numerous awards to their credit. They put on an impeccable show, and I wished they could have done a few more numbers (or the whole night, as far as I was concerned). Also, because they played under just plain white stage light, it was easier for me to get a good clear picture of them.

Del McCoury BandAfter an intermission—a relatively new innovation for the Opry—Lee Greenwood took the stage. He sounded as much like himself as ever. Obviously he had to do “God Bless the USA”; I guess it wouldn’t be a Lee Greenwood performance if he didn’t include that number. Everyone stood and cheered, needless to say.

Lee GreenwoodLee was followed by the Willis Clan. Well, not the entire Willis Clan, of course, but only the majority of the twelve children of the one Willis family. In the first group picture you see the six oldest brothers and sister; another brother was playing the drums in back, and several other siblings made short appearances, as, for example, the latest youngest sister of the bunch, who did a little dance. They played two Irish instrumental pieces; note the rather condensed version of the bag pipe, which—according to them—was exported by the Irish to Scotland and assumed a more inflated nature there. (I think the historic/cultural lines between Ireland and Scotland are a little more intertwined than that.) For their third (and last) number they tipped their proverbial hats to the well-known family act of the past, the von Trapp family, by singing their variation of “My favorite Things” from "The Sound of Music." Their rendition was rather animated and invigorating.

The Willis Clan

The Willis Clan

The Willis Clan

The Willis Clan

Then the star of the night was announced: Keith Urban. The applause and female screeching sounds made it clear that he was not entirely unknown nor without appeal. Let me go on record to say that he was good, and that his backup group was excellent. Let me also add that I have no idea why he has been classified as a country singer. "Urban" is his real name, so there's no intended play on words there. He was born in New Zealand and grew up in Australia, and looks vastly younger than the forty-seven years as of this coming Sunday.  In short, Keith Urban is extremely talented, listenable, and, according to June, "just oozes charm."

Keith UrbanKeith Urban's Bass Player

I know that it’s cool to carry your guitar fairly down on your body, but I’m not sure I’ve ever seen a bass player hold his instrument at a level entirely below his belt. If doing so is more common than is familiar to me, I must not have gone to the right kind of concerts. And I promise that I won't try it.

I’m preparing some further thoughts on this experience in conjunction with whenever I get around to the next conceptual post.

Overall, the concert was good fun, but left me without any further help in answering George Jones’ question.

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Monday, October 20th 2014

23:27

At the Ryman

  • STATE OF EXISTENCE: gruntled

Roy Acuff and Minnie PearlJune and I are once again south of our regular habitat, possibly the last time before the bitter winter hits with its four months of complete darkness, temperatures well below -80°F, and seemingly unending ice and snow.—Oh, wait! I got mixed up there for a moment. We don’t live at the North Pole, even though it may feel that way to me come January. We will have daylight, quite a bit of it compared to what they get 40 degrees of latitude north of our location. The temperatures may go below zero, though hopefully not as much as last winter, and, except for one year long ago (1816), winter has always come to an end in Indiana.

A couple of weeks ago I mentioned that I was going through a time of depression, which was triggered by a number of factors, only one of which was the cancellation of a trip, which would have been more of a professional character. Since we had already moved all appointments to later, the time was there, and we had already set aside some resources. So, June suggested that we could spend a few days in Nashville. Somehow she managed to persuade me that doing so would be a good plan. Well, you know that it was a very easy sell. This time the name "Nashville" refers to the big city in Tennessee, home of the Grand Ole Opry (GOO), in contrast to the small town in Indiana, where the Lil’ Opry has still not been rebuilt, which is our more frequent hide-out.

Today’s small agenda was to make sure that we can find our way to the Ryman Auditorium, which was the location of the GOO from 1942 to 1974. (The GOO is now held at the much larger Opry House, a little further outside of town.) As long as we were there, we took the guided backstage tour, which turned out to be quite interesting, thanks to a tour guide who spoke quite candidly about the various musicians who had played the Opry at the Ryman.

At the Ryman AuditoriumAfterwards, I did the awfully touristy thing of having my picture taken onstage at the microphone. At first I thought that I was being really shallow doing so, but then, when the moment came as the young lady said to go up to the designated place on the stage, it really hit me. Regardless of the contrived set-up, I was about to stand in the same place Johnny Cash, Loretta Lynn, Connie Smith, Roy Acuff, Minnie Pearl, Hank Williams, George Jones, .... we could go on and on ... have stood and played for the crowds. In contrast to other entertainers, so many country singers have not hesitated to let the outside world know of both their wounds and their recovery from them. I certainly wouldn’t use the word “sacred,” but I did have the feeling that there was something special about that space and that it was a privilege to stand there for a moment (at a price, of course). I took one of the guitars provided and sang a few lines of Hank Williams’ “Your Cheatin’ Heart” and then held still for a better shot. I received the picture, but no digital media, so the picture here is my photograph of the commercial photograph. Please forgive me if the impulse of adding an enlarged picture may strike you as a little narcissistic. As I said in my FB post: For a moment or so, a fantasy had come to life, singing a Hank Williams tune on the stage of the Ryman. Kleist would have understood.

There’s a song that keeps popping up in my thoughts. George Jones sang it a couple of decades ago: “Who’s gonna fill their shoes?” Sadly, I don’t know whether anyone is or can. Maybe I’ll be a little more optimistic a few days from now.

A fact that should not be passed over too lightly is that country music and gospel have traditionally gone hand-in-hand. Numerous country musicians have related meaningful testimonies of their faith in Christ, often after they had seen their lives swim down the river of alcohol or similar issues. The Ryman began as a church and has retained that character in its furnishings.  There still are stained-glass windows, and the seating does not consist of plush theater seats, but good old hard wooden pews, as you can still find them in small country churches. My point is very simply that I fear as country music is losing more of its traditional form, that aspect of this musical idiom may get buried as well.

I’ll try to post at least a small entry as often as I can this week. Maybe I’ll get to some more conceptual matters as well, but I’m not promising.

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Thursday, October 16th 2014

21:01

Omega Consistency

  • STATE OF EXISTENCE: okay

Yet another entry that I started several days ago, but hope to finish and post today (Thursday). I began it on Tuesday evening, but I got too tired to finish it. Then yesterday I had The Stomach Ache. That’s right, not a stomach ache, but its real Platonic Form, of which all other stomach aches are but feeble imitations. Okay, I’m totally exaggerating, but it was somewhat unpleasant, and, late in the afternoon, when I realized that the pain wasn’t going vanish by itself, June and I drove to a walk-in clinic [sic], and the wonderful staff there took care of it. And that’s where I shall leave that subject, except to reassure you that it was nothing serious.

*****

Somewhere in our house, I’m quite sure, I have a copy of a biography of Kurt Gödel, the famous logician/mathematician. But I can’t find it. Maybe it will never show up, in which case, I may need to leave the question of whether it is still on the premises undecided. I know that my search has not been complete, but I’m not sure if it will ever be. — If you didn’t get the attempt at humor in the foregoing, don’t worry about it. It wasn’t all that funny. Let me just explain that Kurt Gödel brought up a famous theorem, one part of which stated that every advanced system of logic must contain sentences for which it cannot be determined whether they are true (undecidability), and—the flip side of the coin—that no such system of logic can be used to deduce all possible sentences that it contains (incompleteness).

Speaking of Gödel, the proof for his theorem includes a concept called omega-consistency (hereafter: ω-consistency), and it occurred to me that my search for the book provides a wonderful illustration of this concept. It’s a notion that I find interesting by itself, apart from Gödel’s use of it. Please note, then, that what I’m addressing here is not Gödel’s theorem per se, but an assumption that played a role in his proof of the theorem. To make this illustration work properly, it helps to assume that the house in which June and I live extends to infinity in at least one dimension, whether length, width or height makes no difference, except to the person cleaning it. (If we could afford such a large house, we also could afford outside help.) It is not necessary to assume that the house will exist forever or that I have an infinite life span, though it might make the illustration a little more memorable, and it would stress the significance of the idea behind it.

Let us say that I have good reason to believe that there is a place in my house in which the book is hidden. However, whenever I check any specific place, such as the shelf with logic books in my library or under the living room couch, I do not find the book there. Still, I continue to be convinced that it has to be somewhere in the house. This is ω-consistency. The more general, abstract statement is true; the book is somewhere in the house. But whenever it is applied to specific spots, the statement that the book is there invariably turns out to be false.

In contrast to ω-consistency, “simple" consistency--the kind that we usually have in mind--allows me to assume that on the basis of the statement that the book is somewhere in the house, there is a specific place (we’ll give it the arbitrary name a), where the book is located and that I will eventually find it. Under ω-consistency we cannot make that assumption.

For those who are interested, let me express this idea with a little formal symbolism. Undoubtedly you remember from whenever you studied algebra that the letter x is usually a variable that stands for an “unknown number.” For our purposes, let’s be a little more general and say that variables represent unknown items, including unknown numbers where appropriate. Think of a variable as an empty bucket into which some sort of content can be poured. In formal language, we can say that a variable can be satisfied or instantiated with a constant. A constant can still just be named by a single letter, but now it stands for one and only one specific item. So, here is the simple statement that there is some unknown place where the book is located.

(∃x)Lx

(∃x) stands for “there is”; Lx  means “x is the location of the book.”

There is an “x” such that x is the location of the book.

From there we can substitute a constant for the variable and say that, consequently, there must be a specific place where the book is located and, in order to keep track of this fact, we give this hypothetical place the arbitrary name a. (Note that if I were also looking for my glasses, I could set it up in the same way, but I couldn’t use a again; I would have to use b or another letter because I cannot assume that the book and my glasses must be at the same location. )

When working under the assumption of ω-consistency, things get a little more complex.

Let us continue to insist that we know that the book is somewhere in the house,

(∃x)Lx is true,

but none of the possible locations are panning out for us. Putting it in formal terms, where a, b, c, … n stand for locations in the house,

La is false; Lb is false; Lc is false; … Ln is false.

Here is where it is important that we include infinity. A statement, such as (∃x)Lx cannot be negated by a counterexample, unless we have exhausted all possible candidates for x (its domain), and that’s not possible if the domain is infinite in size. To be sure, it would take only one constant that satisfies the variable x for (∃x)Lx to be true. But ω-consistency is defined as a case in which we cannot find such an instance. By stipulating an infinitely large house (and even if you endowed me with an infinite life span), I could never consider my search to be complete under those restraints.

Consequently, under the stipulation of ω-consistency (∃x)Lx can remain to be true, while La, Lb, Lc, … Ln,  viz. all instantiations of x that are known to us, turn out to be false.

But is it true? It is always a good idea to undertake a reality check, but this question is not the right one to ask here. ω-consistency is a logical or mathematical device, and as such it cannot be true or false, just as multiplication is neither true nor false, but 5x3=15 is true, and 2x6=11 is false. If we are concerned with reality, we need to ask whether it’s applicable. And then the answer is “yes.” An “existential” statement (one preceded by (∃x)—“there is”) that we know to be true on other grounds cannot be falsified by an absence of specific observed instances.

Please let me stick with math a little longer. The set of natural numbers {0, 1, 2, 3 …n} is infinite. Numbers have properties; they may be prime, divisible by two, the square of another number, the cube root of another number, and so forth. For a long list of properties that numbers can have, see Robert Munafo’s Collection. I happened upon that site as I was looking for the details of the example below and found not only what I was looking for, but lots of further information. It's a website written in a personal tone that can easily steal your afternoon if you're not scared of weird numbers.

Anyway, we cannot rule out the possibility that a certain number may have a certain property just because we have not seen that property in any other number before. Some numbers that exhibit certain mathematical functions don’t pop up until we enter a realm so remote that we can write them in scientific notation, but cannot really comprehend them.

So, in that spirit, here’s a mind boggler, at least from where I sit. The Riemann Hypothesis is quite possibly as well-known as it is frequently misrepresented on TV shows. (“The significant zeroes of the zeta function have real value 1/2” just doesn’t emanate the aura of non-Euclidean space. Riemann also made contributions to the latter topic, but that's a different matter.) The famous hypothesis has to do with the distribution and density of prime numbers on the ladder of integers. Regardless of how high up you look, the primes never stop coming, and there is a formula that gets fairly close to predicting where they will appear. This formula asserts that Frank and Stan are roughly of equal height. Well, okay, it’s obviously math, and it's a whole lot more technical; in precise terms it says that the prime-counting function π(x)* is a close approximation of the logarithmic integral function li(x), or π(x) ≈ li(x). However, unless you already know what those things are, we might as well for a moment stick with calling them Frank and Stan, two growing boys, whose height is measured rather frequently. The point is that for a very long time, even though their heights match each other very closely, Frank is almost consistently just a tiny bit shorter than Stan.  π(x) < li(x). But there comes a point when Frank has a growth spurt, and now he is almost always that little fraction taller than Stan. In the actual application, there comes a point at which the ratio is reversed and the prime counting function, π(x), runs a just little larger than the log-integration function, li(x). Sticking with the math now, the question is where on the number line we might find the point that this crossover actually happens. Using the numbers supplied by Munafo, a mathematician named Stanley Skewes established in 1933 that the number could not be any higher than 10101034.  You might find the magnitude of that number highly astounding: a 10 followed by more than 100,000 zeroes. But we now know that Skewes’ estimate was way too high. According to Munafo, “In 2005 numerical techniques were used to determine the actual value of the crossover, 1.397162914×10316. " Of course this number is quite a bit less than Skewes' original guess, but it still is an unimaginable number. We can say it and know what it means, but we cannot “conceive” it or picture any kind of representation of it. Still, the result is meaningful.

Now, if I had been trudging my way along the number line while applying the two functions, I think that I probably would have given up looking for a decisive crossover after the first googol  (10100) or so, quite likely even a whole lot earlier. Yes, undoubtedly much, much earlier if I ever even got started. But other people have had more efficient methods and discovered this switch. So, now we can be pretty certain that,  a change in the relationship between the two functions occurs on the number line roughly around 10316. Why does that happen? We don’t know. Numbers have their own personalities and quirks. They represent one more instance where we see the creativity of their Maker at work, just as much as we can see his hand in the beauty of nature.  

There’s an obvious lesson for the skeptic here. We can put it into easy terms: Don’t be too fast with your conclusions based on what you have not seen. Nietzsche’s famous argument that, given an infinite amount of time, all events in history and in a person’s life must recur over and over again, is flawed because it renders the infinite finite, whereas the set of real numbers can never be exhausted. (Further elaboration on request.) However, the idea that, after working our way along the line of integers for an unimaginable distance and then encountering something that had not been the case before, is real and instantiated. To close briefly, our own few personal, empirical observations cannot possibly be sufficient to rule out the reality of transcendence that we find in God.

*****

Oh, by the way, I found the biography of Gödel a little while ago. It was right where it belonged in the case of logic books. I had looked right at it, but not recognized it. However, we'll leave any further research on that phenomenon for some other time.

*****

*The symbol π does at least double duty in mathematics. Here it is the name of a function in the context of the prime number theorem and has nothing to do with the well-known value of 3.14… in geometry and other areas of math.

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Friday, October 10th 2014

22:40

Zacchaeus in the Tree

  • STATE OF EXISTENCE: holding on

So, there I was this afternoon, drawing a picture of what eventually was supposed to be Zacchaeus in the Sycamore tree. As I was working on making the branches and twigs, it occurred to me all of a sudden that I probably should look up Sycamore trees online, just to make sure that I wasn’t entirely in fantasy land with my intended depiction. I‘ll let you decide how close I came, though I decided I wasn’t going to aim for realism;  I would just draw the tree as I wanted to.

But then another thought came up. Someone who is versed in pseudo-Freudian psychology could analyze my tree for what it reveals concerning my subconscious, perhaps by referring to the “house-tree-person” test and its rules of interpretation. (No, I’m obviouslynot going to link to it.) Oh well, if anyone is so inclined, have at it! Just remember, I didn’t put the little man in the tree; that’s from the Bible story below. If I say so myself, he came out pretty well, given my frugal artistic ability. And if what you see is embarrassing, let me know privately, please!

*****

Moving on to another topic, isn’t the following a common experience? In the last blog entry I wrote about leaving difficult matters in the hands of God. So, the Lord is asking (so to speak), “Do you really mean that, Win?” From my perspective at least, a number of things have crashed down on me over the last few days. Someone else may have seen nothing or perhaps just observed those things come softly wafting down to earth. Regardless, it has felt like crashing to me, thereby triggering a fairly solid bout of depression. I mean, all-in-all, things going wrong shouldn’t make you depressed; they probably should make you sad if you have common reactions. It’s when the sadness takes on a life of its own and puts you into a vicious circle that depression takes over. The apostle Paul found himself in that state several times (1 Thess. 3:5; 2 Cor. 1:8-10). I think I’m in the process of emerging from it, I think, but I also thank you for your prayers.

Luke Bible Study Read the Text!

Bible Reading:
Luke 19:1-10

vv. 9-10: "Today salvation has come to this house,” Jesus told him, “because he too is a son of Abraham. For the Son of Man has come to seek and to save the lost.”

Zacchaeus in the Sycamore TreeObviously there was no way that I could read this passage or write about it without “Zacchaeus was a wee little man” running through my mind. Wait! Why am I saying running? It doesn’t seem to be going anywhere. It’s not so much running as orbiting through my brain.

Anyway, it’s a good little song, and it’s a great story that culminates with Jesus declaring that he has come “to seek and to save the lost.” That would be those who are rejected and considered dispensable, the ones who know how far they have fallen short of God’s expectations. Nevertheless, Jesus seeks them out.  

We can contrast this sentence with Paul’s description of our fallenness before we come to Christ: There is no one who understands; there is no one who seeks God (Rom. 3:11). Regardless of whether we consider ourselves Arminians, Wesleyans, or Calvinists, we all believe that God made the first move and sought us out to become his children.

Just as hard as it is for me to reflect on this story without humming “Zacchaeus,” I also find it impossible to avoid the glaring contrast between this episode and the one in the last chapter concerning the so-called rich young ruler. Or, for that matter, not to tie this story into the parable of the two men praying in the temple. I don’t think that it’s unrealistic to believe that Luke placed those narratives in close proximity so that we would see the differences. The ruler of the synagogue in the last chapter was among the religious elite of his day. Zacchaeus, on the other hand, was considered pond scum. The tax collectors of the day were working for the pagan Romans who had occupied the land. And, if that wasn’t enough, they were also well known for their dishonesty. They were obligated to keep a schedule according to which they would turn over a certain sum of money to the Romans. They had to raise these funds from the people, and they were entitled to increase the amount somewhat, with the proceeds constituting their income. Apparently, it was all-too-easy for tax collectors to increase their net gain with impunity and become quite rich in the process.

Zacchaeus had worked his way up the official ladder, being designated a chief tax collector, and quite a bit of money had gathered in his coffers. It seems pretty clear to me, however, that he did not feel entirely good about his life. Obviously, there are some interesting aspects to this story on a purely superficial level. Jesus, the celebrity, was walking through Jericho, and Zacchaeus, being of small stature, climbed a tree in order to get a look at Jesus over other peoples’ heads.

But why was he so desperate to see Jesus? I think the answer is given by how quick he was to welcome Jesus into his house and to use the occasion to repent. I don’t think that we’re going out on a limb by inferring that the man was guilt-ridden. (Oh no, I just recognized the horrible pun in the process of proof-reading. I’ll leave it stand, though.) Zacchaeus would have been familiar with Jesus in his role as a popular healer. But then Jesus turned to him personally and invited himself to Zacchaeus’ house, thereby doing him a great honor according to Middle Eastern customs to this day, he realized that he had found the person who could make him righteous again.

He had wanted to see Jesus. We don’t know whether he expected Jesus to see him, or whether he expected anyone among the crowd to notice him sitting on a branch of the tree or, for that matter, to care. But Jesus did see him and not only called him down, but simultaneously stated that he would spend the night at the house of Zacchaeus. (Apparently nobody had made reservations for Jesus at a place in Jericho. That reminds me of an adventure in Jericho a number of years ago, which I may have already written about some other time, but, if not, it doesn’t fit in here particularly.)

The crowd went into shock, both individually and collectively. They had been parading along with Jesus cheering him on. There was that little unpleasantness with the blind man, but that had turned out great because Jesus made a miracle, as everyone had hoped. But now he had just said that he would take his lodgings with a—it’s hard to believe, but there’s no getting around it—a—could this really be true—a tax collector. Woosh! A sudden chill swept through the air. Jesus was going to stay with a genuine, certified sinner.

Well, sensitivity to public opinion would not have been an asset to anyone pursuing the career of a tax collector, and Zacchaeus didn’t seem pay attention to this reaction. He came down from the tree as a happy man, and he declared to Jesus that he would make good on any wrongful transactions as well as give half of his possessions to the poor. —Ah, wasn’t the price for salvation all of your possessions?— No, as I explained, that wasn’t the point when Jesus stated the matter to the synagogue official. Salvation can’t be bought. The idea was to show unreserved commitment to God, not the exact sum you would give to charity.

Jesus saw that Zacchaeus’ conversion was real. Partially acknowledging Zacchaeus, and perhaps also partially in rebuke of the crowd, many of whom would still have been in earshot, he declared: “Salvation has come to this house.” And, just to make it clear, he added that “he, too, is a son of Abraham.” Ouch! How that must have stung! We have seen how folks like the rich official took great pride in following the Law of God, thereby cashing in on their privilege of being one of Abraham’s descendants. But Zacchaeus had the same ancestor. Remember that we are still at a time prior to Pentecost, and the mystery of salvation for the Gentiles as Gentiles had not yet become disclosed, though Jesus was frequently hinting at it. But here he confined himself to the current context, making a sufficiently controversial statement to set the crowd reeling. “He too is a son of Abraham.”

What the people did not yet know would become evident not long thereafter. “[Abraham] is the father of us all in God’s sight!” Paul declaims in Rom. 4:16b, 17a; see also Gal. 4:14. And, if the division between Jews and Gentiles does not interfere with God’s acceptance of you, neither does your state as a sinner, as shown by Zacchaeus. No one is so far from God that he or she is beyond his reach.

 

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Sunday, October 5th 2014

20:09

A Blind Man's Faith

  • STATE OF EXISTENCE: George McFly-like

George McFlyFeeling fairly good physically. No headache for three days now. Of course, now I have to go get my shingles shot, but I haven't heard from anyone that they have reacted to it. Come to think of it, I know a number of people who have had shingles, but actually I'm not sure I know anyone of whom I could say with certainty that they had the shingles immunization. Presumably medical personnel would have had a shingles shot, but no one has told me. I am aware of the fact that the shingles vaccine does not totally prevent everyone who has received it from getting shingles; it's supposed to lower the probability and, I guess, the intensity. If anything dramatic should occur, I'll let you know.

It’s definitely October. For a number of years we’ve managed to keep away from October for a while by escaping to warmer weather, only to return to take it on the chin by November. This year, if we still do wind up heading anywhere this month, the distance and length will be much shorter than, say several weeks in South East Asia. I’ll tell you more once I’m sure that something has come together. Unfortunately some other plans have fallen apart.

Luke Bible Study Read the Text!

Bible Reading:
Luke 18:35-43

v. 38: So he called out, “Jesus, Son of David, have mercy on me!”

Travel map Galilee to JerusalemWhen Jesus made his occasional trips from Galilee to Jerusalem, he was likely to go through Samaria, the direct and shorter route. Most Jews at the time avoided Samaria and took a much longer trip by crossing the Jordan, trekking for a short time through the area called Decapolis, traversing Perea southward, crossing the Jordan somewhere in the vicinity of Jericho, and finally completing the last leg of their journey by walking uphill to Jerusalem. Decapolis was a loose federation of ten (more or less) cities that were relatively independent of each other, though under the strict supervision of Rome. The one thing that they had in common was that in various ways they promoted Hellenistic Greek and Roman culture and religion. Perea, on the other hand, was nominally Jewish during the time of Jesus. It was the name of the region located on the East Bank of the Jordan, governed by the Roman puppets Herod the Great (the baby killer) and subsequently by his son, the tetrarch Herod Antipas. Many centuries earlier, the regions that were then called Decapolis and Perea had been home to the tribes of Reuben, Gad, and half of Manasseh, but way back in 732 BC the Assyrian king Tiglath-pileser (744-727 BC) had carried them off to captivity. Nowadays, most of Decapolis/Perea belongs to the Kingdom of Jordan and a snippet of it goes to Syria.

On this last trip, Jesus chose to follow this route, and his entry into Jericho was a big event for the city. There was a large crowd surrounding him, presumably shouting their welcomes and accolades to him. A blind man, sitting just a little out of the way, inquired what was going on. Someone gave him a quick answer. “Jesus the Nazarene is passing by.”

Please don’t confuse “Nazarene” with “Nazarite.” The latter would have been a man during Old Testament times who had taken a temporary vow of holiness and would abstain from having his hair cut, drinking beer, or consuming anything connected to grapes, including table grapes, juice, or wine. “Nazarene” simply refers to Jesus’ original home city of Nazareth, identifying him as a Galilean. Here in Judea this distinction was important to the population who at times thought of themselves as just a little better than their relatives to the north.

Apparently the blind man understood something that the rest of the crowed missed. He wanted Jesus to cure his blindness, but he did not address Jesus with the title someone had just mentioned to him. He did not shout, “Jesus of Nazareth, have mercy on me!” His call was “Jesus, Son of David, have mercy on me!” The term, Son of David, was one of the many with messianic implications. So, while the rest of the crowd was excited that Jesus the Nazarene, well-known preacher and healer—perhaps best known for his feeding miracles—had come to Jericho, the blind man had the “insight” that the messiah had come. And so he kept calling “Son of David, have mercy on me!”

A number of people got irritated with the blind man; perhaps they thought that he was spoiling the party. Not that they would have objected to witnessing a solid miracle or two, but this blind man was pushing his own problems just a little too much and was creating a racket that others found annoying. “Alright already! Everybody has heard you—lots of times. But your behavior is taking the joy out of it for others. Would you please quiet down; we can hardly hear what the Nazarene is saying. You’re not the only person here who needs him.”

Regardless of how the people admonished him, the blind man did not let them deter him. He just kept calling for the “Son of David” to show him his mercy.

It is easy to lose track of individuals in a large crowd, but it seems that Jesus was frequently aware of special people in the throng around him. In this case, that point was obviously coupled with the fact that his unceasing cry for help was not easy to miss. Consequently, Jesus called him over, and some people led him into his presence.

“So, tell me, what exactly do you want me to do for you?” Jesus asked. After all, “have mercy on me” could have referred to any number of items.

The astute blind man clearly did not mince any words. “I want to see!” No qualifications, no false humility, no attempts at bargaining, as in “If you will give me sight, I promise to …” No suggestions to Jesus how he should do so. Just plain and simple, “I want to see.” The man was convinced that a) he had a problem, namely blindness, and b) that Jesus could fix it. Now that he had Jesus’ attention, he could just leave the matter in his hands.

I use a phrase along the line of “leaving matters in God’s hands” quite frequently on this blog. By now, it almost writes itself. And yet, what a difficult thing to put these words into action (or, perhaps better into non-action): to turn something over to Jesus and then let him decide if, when, and how to do something about it.

So many times, we want to give God incentive. “If you do this we will praise you from the roof tops, and so forth.” Aren’t we supposed to do that already, at least figuratively?

“If you’ll answer my prayer, God, I promise that I will …” If God is pleased by your doing X, then you should be doing X already.”

At other times, we make suggestions to God how he could answer a prayer. “Lord, please guide the surgeon’s hand,” and we might as well add, “help the anesthesiologist get the right dosages, and help the nurses to keep track of drainages, etc.” God knows what's involved and doesn't need our specific guideline. But it's just fine if you want to tell him about all the issues that could go wrong.

So, I'm not giving a criticism here. God is not offended by our prayers, even if they reflect a naïve theology. You can’t surprise God with your prayers; he already knows what’s going on in your heart and mind. Please, I beg you, pray what you’re thinking, even if your prayers seem as silly to you as mine do to me at times. It’s a wonderful experience when you learn to be honest with yourself and God. (Which one comes first?, I wonder.)

What I’m talking about only becomes a problem if we get the idea into our heads that our attempts to “persuade” God rather than just unburdening ourselves to him are more likely to produce “results.” God does not work that way. What Jesus saw in the blind man was unwavering dependence on him, not a better technique or religiosity. The man recognized that Jesus was the messiah, and that, if anyone could take his darkness away, it would be this Son of David. And Jesus told him, “Your faith has healed you.”

Let me quickly emphasize that Jesus was not giving a full elaboration on prayer and faith here. “Faith” by itself does nothing. In fact, “faith” by itself is nothing. “Faith” is only something when it is either “faith that …” or “faith in …” So, Christ’s statement fully expressed to my theological satisfaction (which undoubtedly was of no concern whatsoever to him then and is not now) would have been. “Your faith in me as the one who will have mercy on you has healed you.” I’m making a deal out of this distinction only because the world is full of people who celebrate “faith” as though it were a thing in itself that in some mystical way does things for us. They believe that even if you don’t believe in God or in false gods, if you have “faith” that’s all it takes. Jesus was saying nothing of the kind.

The blind man was healed, and glorified God.  The crowd was excited, and—maybe taking their cue from this man—praised God.

Now I’m going to read something into the text that’s not there; it’s a hypothesis based only on my picture of the blind man and his explicit faith, which Jesus commended. What would he have done if for whatever reason he would not have received his sight at that point. My feeling (and it’s nothing more than that) is that he very well may have sat down again at his regular place and said to himself. “Okay, maybe next time.” That’s the position we often find ourselves in, and it’s not easy to continue to have unreserved faith in God. Then again, there’s no one bigger or better to have faith in than the Creator and Sustainer of the universe.

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